QCD Critical Point : Inching Towards Continuum

Transcription

1 QCD Critical Point : Inching Towards Continuum Rajiv V. Gavai T. I. F. R., Mumbai, India Introduction Lattice QCD Results Searching Experimentally Summary Work done with Saumen Datta & Sourendu Gupta New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 1

2 Introduction QCD Critical Point in T -µ B plane A fundamental aspect; New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 2

3 Introduction QCD Critical Point in T -µ B plane A fundamental aspect; Based on symmetries and models, the Expected QCD Phase Diagram From Rajagopal-Wilczek Review New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 2

4 Introduction QCD Critical Point in T -µ B plane A fundamental aspect; Based on symmetries and models, the Expected QCD Phase Diagram... but could, however, be... Τ From Rajagopal-Wilczek Review New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 2

5 Introduction QCD Critical Point in T -µ B plane A fundamental aspect; Based on symmetries and models, the Expected QCD Phase Diagram... but could, however, be... Pisarski 2007; Castorina-RVG-Satz 2010) (McLerran- Τ From Rajagopal-Wilczek Review Constituent Q-Gas (PC-RVG-Satz) New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 2

6 The Race is ON Between the theorists, to claim a patch on the QCD phase diagram, Between the model builders and the lattice folks for precise location, Between the phenomenologists to come up with the best tool to find it; New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 3

7 The Race is ON Between the theorists, to claim a patch on the QCD phase diagram, Between the model builders and the lattice folks for precise location, Between the phenomenologists to come up with the best tool to find it; Between the theorists and experimentalists : to establish/locate OR to disprove; And, of course, between the various ongoing (RHIC/STAR) and the proposed/designed experiments (FAIR/CBM). New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 3

8 The Race is ON Between the theorists, to claim a patch on the QCD phase diagram, Between the model builders and the lattice folks for precise location, Between the phenomenologists to come up with the best tool to find it; Between the theorists and experimentalists : to establish/locate OR to disprove; And, of course, between the various ongoing (RHIC/STAR) and the proposed/designed experiments (FAIR/CBM). Maybe it is the synergy between one or more of them that will eventually lead us to the holy grail. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 3

9 Lattice QCD Results Lattice QCD Most Reliable and Completely parameter-free way to extract non-perturbative physics relevant to Heavy Ion Colliders. The Transition Temperature T c, the Equation of State (used now in elliptic flow analysis), and the Wróblewski Parameter λ s etc. (Wuppertal-Budapest, HotQCD, GG 02) New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 4

10 Lattice QCD Results Lattice QCD Most Reliable and Completely parameter-free way to extract non-perturbative physics relevant to Heavy Ion Colliders. The Transition Temperature T c, the Equation of State (used now in elliptic flow analysis), and the Wróblewski Parameter λ s etc. (Wuppertal-Budapest, HotQCD, GG 02) Flavour Correlations (C BS ) and Charm Diffusion Coefficient D are some more such examples for RHIC Physics. (Gavai-Gupta, PRD 2006 & Banerjee et al. PRD 2012) X=Q * LOPT XS C π DT 16 8 X=B T/Tc 0 PHENIX T/T c New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 4

11 The µ 0 problem : Quark Type Mostly staggered quarks used in these simulations. Broken flavour and spin symmetry on lattice. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 5

12 The µ 0 problem : Quark Type Mostly staggered quarks used in these simulations. Broken flavour and spin symmetry on lattice. Domain Wall or Overlap Fermions better. BUT Computationally expensive. Introduction of µ a la Bloch & Wettig (PRL 2006 & PRD2007) New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 5

13 The µ 0 problem : Quark Type Mostly staggered quarks used in these simulations. Broken flavour and spin symmetry on lattice. Domain Wall or Overlap Fermions better. BUT Computationally expensive. Introduction of µ a la Bloch & Wettig (PRL 2006 & PRD2007) Unfortunately breaks chiral symmetry! (Banerjee, Gavai & Sharma PRD 2008; PoS (Lattice 2008); PRD 2009 ) Good News : Problem Solved! Overlap Lattice Action with exact chiral invariance at nonzero µ and any a now exists (Gavai & Sharma, arxiv : ; PLB in press, Narayanan-Sharma JHEP 11). New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 5

14 Using chiral projectors for overlap fermions as, ψ L = [1 γ 5 (1 ad ov )]ψ/2 & ψ R = [1 + γ 5 (1 ad ov )]ψ/2, leaving the antiquark field decomposition as in the continuum, the overlap action for nonzero µ is S F = n [ ψ n,l (ad ov + aµγ 4 )ψ n,l + ψ n,r (ad ov + aµγ 4 )ψ n,r ] = n ψ n [ad ov + aµγ 4 (1 ad ov /2)]ψ n. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 6

15 Using chiral projectors for overlap fermions as, ψ L = [1 γ 5 (1 ad ov )]ψ/2 & ψ R = [1 + γ 5 (1 ad ov )]ψ/2, leaving the antiquark field decomposition as in the continuum, the overlap action for nonzero µ is S F = n [ ψ n,l (ad ov + aµγ 4 )ψ n,l + ψ n,r (ad ov + aµγ 4 )ψ n,r ] = n ψ n [ad ov + aµγ 4 (1 ad ov /2)]ψ n. Easy to check that under the chiral transformations, δψ = iαγ 5 (1 ad ov )ψ and δ ψ = iα ψγ 5, it is invariant or all values of aµ and a. Order parameter exists for all µ and T. It is ψψ = lim am 0 lim V Tr (1 ad ov /2) [ad ov +(am+aµγ 4 )(1 ad ov /2)]. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 6

16 The µ 0 problem : The Measure Simulations can be done IF Det M > 0. However, det M is a complex number for any µ 0 : The Phase/sign problem New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 7

17 The µ 0 problem : The Measure Simulations can be done IF Det M > 0. However, det M is a complex number for any µ 0 : The Phase/sign problem Several Approaches proposed in the past two decades : Two parameter Re-weighting (Z. Fodor & S. Katz, JHEP 0203 (2002) 014 ). Imaginary Chemical Potential (Ph. de Forcrand & O. Philipsen, NP B642 (2002) 290; M.-P. Lombardo & M. D Elia PR D67 (2003) ). Taylor Expansion (C. Allton et al., PR D66 (2002) & D68 (2003) ; R.V. Gavai and S. Gupta, PR D68 (2003) ). Canonical Ensemble (K. -F. Liu, IJMP B16 (2002) 2017, S. Kratochvila and P. de Forcrand, Pos LAT2005 (2006) 167.) Complex Langevin (G. Aarts and I. O. Stamatescu, arxiv: and its references for earlier work ). New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 7

18 Why Taylor series expansion? Ease of taking continuum and thermodynamic limit. Better control of systematic errors. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 8

19 Why Taylor series expansion? Ease of taking continuum and thermodynamic limit. Better control of systematic errors. E.g., exp[ S] factor makes this exponentially tough for re-weighting. Discretization errors propagate in an unknown manner in re-weighting. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 8

20 Why Taylor series expansion? Ease of taking continuum and thermodynamic limit. Better control of systematic errors. T E.g., exp[ S] factor makes this exponentially tough for re-weighting. Discretization errors propagate in an unknown manner in re-weighting. V 2 V1 µ We studied volume dependence at several T to i) bracket the critical region and then ii) tracked its change as a function of volume. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 8

21 Details of Expansion Standard definitions yield various number densities and susceptibilities : n i = T V ln Z µ i and χ ij = T V 2 ln Z µ i µ j. These are also useful by themselves both theoretically and for Heavy Ion Physics (Flavour correlations, λ s...) Denoting higher order susceptibilities by χ nu,n d, the pressure P has the expansion in µ: P T 4 P (µ, T ) T 4 P (0, T ) T 4 = 1 χ nu,n d n n u,n u! d ( µu T ) nu 1 n d! ( µd T ) nd New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 9

22 Details of Expansion Standard definitions yield various number densities and susceptibilities : n i = T V ln Z µ i and χ ij = T V 2 ln Z µ i µ j. These are also useful by themselves both theoretically and for Heavy Ion Physics (Flavour correlations, λ s...) Denoting higher order susceptibilities by χ nu,n d, the pressure P has the expansion in µ: P T 4 P (µ, T ) T 4 P (0, T ) T 4 = 1 χ nu,n d n n u,n u! d ( µu T ) nu 1 n d! ( µd T ) nd New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 9

23 From this expansion, a series for baryonic susceptibility can be constructed. Its radius of convergence gives estimate of the location of nearest critical point. Successive estimates for the radius of convergence obtained from these using ( ) n(n+1)χ (n+1) χ B or n! (2) 1/n B. We use both and terms up to 8th order in µ. χ (n+3) B T 2 χ (n+2) B T 2 All coefficients of the series must be POSITIVE for the critical point to be at real µ, and thus physical. We (Gavai-Gupta 05, 09) use up to 8 th order. B-RBC so far has up to 6 th order. 10th & even 12th order may be possible : Ideas to extend to higher orders are emerging (Gavai-Sharma PRD 2012 & PRD 2010) which save up to 60 % computer time. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 10

24 From this expansion, a series for baryonic susceptibility can be constructed. Its radius of convergence gives estimate of the location of nearest critical point. Successive estimates for the radius of convergence obtained from these using ( ) n(n+1)χ (n+1) χ B or n! (2) 1/n B. We use both and terms up to 8th order in µ. χ (n+3) B T 2 χ (n+2) B T 2 All coefficients of the series must be POSITIVE for the critical point to be at real µ, and thus physical. We (Gavai-Gupta 05, 09) use up to 8 th order. B-RBC so far has up to 6 th order. 10th & even 12th order may be possible : Ideas to extend to higher orders are emerging (Gavai-Sharma PRD 2012 & PRD 2010) which save up to 60 % computer time. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 10

25 From this expansion, a series for baryonic susceptibility can be constructed. Its radius of convergence gives estimate of the location of nearest critical point. Successive estimates for the radius of convergence obtained from these using ( ) n(n+1)χ (n+1) χ B or n! (2) 1/n B. We use both and terms up to 8th order in µ. χ (n+3) B T 2 χ (n+2) B T 2 All coefficients of the series must be POSITIVE for the critical point to be at real µ, and thus physical. We (Gavai-Gupta 05, 09) use up to 8 th order. B-RBC so far has up to 6 th order. 10th & even 12th order may be possible : Ideas to extend to higher orders are emerging (Gavai-Sharma PRD 2012 & PRD 2010) which save up to 60 % computer time. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 10

26 The Susceptibilities All susceptibilities can be written as traces of products of M 1 and various derivatives of M. At leading order, χ 20 = ( T V ) [ O 2 + O 11 ], χ 11 = ( T V ) [ O 11 ] Here O 2 = Tr M 1 M Tr M 1 M M 1 M, and O 11 = (Tr M 1 M ) 2, and the traces are estimated by a stochastic method (Gottlieb et al., PRL 87): Tr A = N v i=1 R i AR i/2n v, and (Tr A) 2 = 2 L i>j=1 (Tr A) i(tr A) j /L(L 1), where R i is a complex vector from a set of N v subdivided in L independent sets. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 11

27 The Susceptibilities All susceptibilities can be written as traces of products of M 1 and various derivatives of M. At leading order, χ 20 = ( T V ) [ O 2 + O 11 ], χ 11 = ( T V ) [ O 11 ] Here O 2 = Tr M 1 M Tr M 1 M M 1 M, and O 11 = (Tr M 1 M ) 2, and the traces are estimated by a stochastic method (Gottlieb et al., PRL 87): Tr A = N v i=1 R i AR i/2n v, and (Tr A) 2 = 2 L i>j=1 (Tr A) i(tr A) j /L(L 1), where R i is a complex vector from a set of N v subdivided in L independent sets. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 11

28 Higher order NLS are more involved. E.g., at the 8th order, terms involve operators up to O 8 which in turn have terms up to 8 quark propagators and combinations of M and M. In fact, the entire evaluation of the χ 80 needs 20 inversions of Dirac matrix. This can be reduced to 8 inversions using an action linear in µ (Gavai-Sharma PRD 2012 & PRD 2010), leading still to results in agreement with that exponential in µ. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 12

29 Higher order NLS are more involved. E.g., at the 8th order, terms involve operators up to O 8 which in turn have terms up to 8 quark propagators and combinations of M and M. In fact, the entire evaluation of the χ 80 needs 20 inversions of Dirac matrix. This can be reduced to 8 inversions using an action linear in µ (Gavai-Sharma PRD 2012 & PRD 2010), leading still to results in agreement with that exponential in µ. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 12

30 Our Simulations & Results Staggered fermions with N f = 2 of m/t c = 0.1; R-algorithm used. m π /m ρ = 0.31 ± 0.01 (MILC); Kept the same as a 0 (on all N t ). Earlier Lattice : 4 N 3 s, N s = 8, 10, 12, 16, 24 (Gavai-Gupta, PRD 2005) Finer Lattice : 6 N 3 s, N s = 12, 18, 24 (Gavai-Gupta, PRD 2009). New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 13

31 Our Simulations & Results Staggered fermions with N f = 2 of m/t c = 0.1; R-algorithm used. m π /m ρ = 0.31 ± 0.01 (MILC); Kept the same as a 0 (on all N t ). Earlier Lattice : 4 N 3 s, N s = 8, 10, 12, 16, 24 (Gavai-Gupta, PRD 2005) Finer Lattice : 6 N 3 s, N s = 12, 18, 24 (Gavai-Gupta, PRD 2009). Even finer Lattice : This Talk (Datta-RVG-Gupta, 12) Aspect ratio, N s /N t, maintained four to reduce finite volume effects. Simulations made at T/T c = 0.90, 0.92, 0.94, 0.96, 0.98, 1.00, 1.02, 1.12, 1.5 and Typical stat in max autocorrelation units. T c defined by the peak of Polyakov loop susceptibility. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 13

32 µ/(3τ) T/Tc= n New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 14

33 µ/(3τ) T/Tc=0.99 T/Tc= n New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 14

34 µ/(3τ) T/Tc=0.99 T/Tc=0.97 T/Tc= n New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 14

35 µ/(3τ) T/Tc=0.99 T/Tc=0.97 T/Tc= n T E T c = 0.94 ± 0.01, and µe B = 1.8 ± 0.1 for finer lattice: Our earlier coarser T E lattice result was µ E B /T E = 1.3 ± 0.3. Infinite volume result: to 1.1(1) Critical point at µ B /T 1 2. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 14

36 Cross Check on µ E /T E Use Padé approximants for the series to estimate the radius of convergence. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 15

37 Cross Check on µ E /T E Use Padé approximants for the series to estimate the radius of convergence. 5 P χ Β /Τ P µ/τ Consistent Window with our other estimates. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 15

38 Critical Point : Story thus far T/Tc GeV 20 GeV 18 GeV (CERN) 10 GeV 0.8 Freezeout curve µ B /T N f = 2 (magenta) and 2+1 (blue) (Fodor-Katz, JHEP 04). N t = 4 Circles (GG 05 & Fodor-Katz JHEP 02), N t = 6 Box (GG 09). New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 16

39 χ 2 for N t = 8, 6, and 4 lattices /T 2 χ Nt=4 Nt=6 Nt=8 PRELIMINARY T/Tc N t = 8 (Datta-Gavai-Gupta, QM12) and 6 (GG, PRD 09) results agree. β c (N t = 8) agrees with Gottlieb et al. PR D47,1993. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 17

40 χ n B χ n B for N t = 8 lattice n=2 n=4 n=6 PRELIMINARY T/Tc 100 configurations & 1000 vectors at each point employed. More statistics coming in critical region. Window of positivity in anticipated region. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 18

41 Radius of Convergence result /T µ B crit Nt=6 Nt=8 PRELIMINARY 2/4 2/6 2/8 4/6 6/8 obtained from ratios of orders At our (T E, µ E ) for N t = 6, the ratios display constancy for N t = 8 as well. Currently : Similar results at neighbouring T/T c = a larger T at same µ E B. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 19

42 Critical Point : Inching Towards Continuum 1.1 Critical point estimates: T/Tc GeV Budapest-Wuppertal Nt=4 Mumbai Nt=8 Mumbai Nt=6 Mumbai Nt= GeV Freezeout curve µ B /T New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 20

43 Lattice predictions along the freezeout curve Hadron yields well described using Statistical Models, leading to a freezeout curve in the T -µ B plane. (Andronic, Braun-Munzinger & Stachel, PLB 2009 ; Oeschler, Cleymans, Redlich & Wheaton, 2009) dn/dy 10 2 s NN =200 GeV Data π + STAR PHENIX BRAHMS Model, χ 2 /N df =29.7/11 3 T=164 MeV, µ = 30 MeV, V=1950 fm b + π K K p p Λ Λ Ξ + Ξ Ω φ d d K* Σ * Λ* New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 21

44 Lattice predictions along the freezeout curve Hadron yields well described using Statistical Models, leading to a freezeout curve in the T -µ B plane. (Andronic, Braun-Munzinger & Stachel, PLB 2009 ; Oeschler, Cleymans, Redlich & Wheaton, 2009) 0.2 dn/dy 10 2 s NN =200 GeV T (GeV) SIS AGS SPS RHIC Data π + STAR PHENIX BRAHMS Model, χ 2 /N df =29.7/11 3 T=164 MeV, µ = 30 MeV, V=1950 fm b + π K K p p Λ Λ Ξ + Ξ Ω φ d d K* Σ * Λ* \/s NN (GeV) µ B (GeV) New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 21

45 Lattice predictions along the freezeout curve Hadron yields well described using Statistical Models, leading to a freezeout curve in the T -µ B plane. (Andronic, Braun-Munzinger & Stachel, PLB 2009 ; Oeschler, Cleymans, Redlich & Wheaton, 2009) 0.2 dn/dy 10 2 s NN =200 GeV T (GeV) SIS AGS SPS RHIC Data π + STAR PHENIX BRAHMS Model, χ 2 /N df =29.7/11 3 T=164 MeV, µ = 30 MeV, V=1950 fm b + π K K p p Λ Λ Ξ + Ξ Ω φ d d K* Σ * Λ* \/s NN (GeV) Our Key Proposal : Use the freezeout curve from hadron abundances to predict fluctuations using lattice QCD along it. (Gavai-Gupta, TIFR/TH/10-01, arxiv ) µ B (GeV) New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 21

46 T/Tc t h coexistence curve µ B/T New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 22

47 T/Tc t h coexistence curve freezeout curve µ B/T New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 23

48 T/Tc t h coexistence curve freezeout curve µ B/T New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 24

49 T/Tc t h coexistence curve freezeout curve µ B/T New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 25

50 T/Tc t h coexistence curve freezeout curve µ B/T Use the freezeout curve to relate (T, µ B )to s and employ lattice QCD predictions along it. (Gavai-Gupta, TIFR/TH/10-01, arxiv ) New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 25

51 T/Tc t h coexistence curve freezeout curve µ B/T Use the freezeout curve to relate (T, µ B )to s and employ lattice QCD predictions along it. (Gavai-Gupta, TIFR/TH/10-01, arxiv ) Define m 1 = T χ(3) (T,µ B ), m χ (2) (T,µ B ) 3 = T χ(4) (T,µB ), and m χ (3) (T,µ B ) 2 = m 1 m 3 and use the Padè method to construct them. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 25

52 10 1 Nt=4 Nt= Nt=4 Nt=6 m m S (GeV) NN S (GeV) NN 1000 Nt=4 Nt=6 100 m Filled circles: negative values S (GeV) NN Gavai & Gupta, arxiv: New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 26

53 HRG power law 4 3 m m HRG S (GeV) NN S (GeV) NN m HRG power law Filled circles: negative values S (GeV) NN Used T c (µ = 0) = 170 MeV (Gavai & Gupta, arxiv: ). New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 27

54 m m Tc(P)=170 MeV Nt=8 6 Resonance gas S (GeV) Tc(P)=170 MeV Nt= S (GeV) 100 filled symbols: negative m 3 10 m Tc(P)=170 MeV Nt= 8 6 Power law S (GeV) 1000 Marginal change if T c = 175 MeV (Datta, Gavai & Gupta, QM 12). New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 28

55 Gavai-Gupta, 10 & Datta-Gavai-Gupta, QM 12 New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 29

56 Smooth & monotonic behaviour for large s : m 1 and m 3. Note that even in this smooth region, an experimental comparison is exciting : Direct Non-Perturbative test of QCD in hot and dense environment. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 30

57 Smooth & monotonic behaviour for large s : m 1 and m 3. Note that even in this smooth region, an experimental comparison is exciting : Direct Non-Perturbative test of QCD in hot and dense environment. Our estimated critical point suggests non-monotonic behaviour in all m i, which should be accessible to the low energy scan of RHIC BNL! New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 30

58 Smooth & monotonic behaviour for large s : m 1 and m 3. Note that even in this smooth region, an experimental comparison is exciting : Direct Non-Perturbative test of QCD in hot and dense environment. Our estimated critical point suggests non-monotonic behaviour in all m i, which should be accessible to the low energy scan of RHIC BNL! Proton number fluctuations (Hatta-Stephenov, PRL 2003) Neat idea : directly linked to the baryonic susceptibility which ought to diverge at the critical point. Since diverging ξ is linked to σ mode, which cannot mix with any isospin modes, expect χ I to be regular. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 30

59 Smooth & monotonic behaviour for large s : m 1 and m 3. Note that even in this smooth region, an experimental comparison is exciting : Direct Non-Perturbative test of QCD in hot and dense environment. Our estimated critical point suggests non-monotonic behaviour in all m i, which should be accessible to the low energy scan of RHIC BNL! Proton number fluctuations (Hatta-Stephenov, PRL 2003) Neat idea : directly linked to the baryonic susceptibility which ought to diverge at the critical point. Since diverging ξ is linked to σ mode, which cannot mix with any isospin modes, expect χ I to be regular. Leads to a ratio χ Q :χ I :χ B = 1:0:4 Assuming protons, neutrons, pions to dominate, both χ Q and χ B can be shown to be proton number fluctuations only. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 30

60 Aggarwal et al., STAR Collaboration, arxiv : Reasonable agreement with our lattice results. Where is the critical point? New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 31

61 Xiaofeng Luo, QM 12 Gavai-Gupta, 10 From STAR Collaboration Datta-Gavai-Gupta, QM 12 New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 32

62 Summary Phase diagram in T µ has begun to emerge: Different methods, similar qualitative picture. Critical Point at µ B /T 1 2. Our results for N t = 8 first to begin the inching towards continuum limit. New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 33

63 Summary Phase diagram in T µ has begun to emerge: Different methods, similar qualitative picture. Critical Point at µ B /T 1 2. Our results for N t = 8 first to begin the inching towards continuum limit. T/Tc GeV Critical point estimates: Budapest-Wuppertal Nt=4 Mumbai Nt=8 Mumbai Nt=6 Mumbai Nt=4 Critical Point leads to structures in m i on the Freeze-Out Curve. STAR results appear to agree with our Lattice QCD predictions. 10 GeV 0.8 Freezeout curve µ B /T New Frontiers in Lattice Gauge Theory, Galileo Galilei Institute, Florence, Italy, August 31, 2012 R. V. Gavai Top 33